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\def\prp{\perp}
\def\Prp{T}
\def\sx{small-$x$}
\def\kt{\ensuremath{k_\prp}}
\def\kti#1{\ensuremath{k_{\prp #1}}}
\def\pt{\ensuremath{p_\prp}}
\def\pti#1{\ensuremath{p_{\prp #1}}}
\def\qt{\ensuremath{q_\prp}}
\def\qti#1{\ensuremath{q_{\prp #1}}}
\def\xbj{\ensuremath{x_{Bj}}}
\def\xjet{\ensuremath{x_{jet}}}
\def\ptjet{\ensuremath{p_{t\;jet}}}
\newcommand{\dg}{\ensuremath{^{\circ}}}
\def\DJANGO{{\sc Django}}
\def\RAPGAP{{\sc Rapgap}}
\def\CASCADE{{\sc Cascade}}
\def\ARIADNE{{\sc Ariadne}}
\def\ldcmc{{\small LDCMC}}
\def\PHOJET{{\sc Phojet}}
\def\HERACLES{{\sc Heracles}}
\def\DISENT{{\sc Disent}}
\newcommand{\gap}{\stackrel{>}{\sim}}
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\newcommand{\LEPTO}{Ingelman_LEPTO65}
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\newcommand{\DJANGOMC}{DJANGO}
\newcommand{\HERACLESMC}{HERACLESa,HERACLESb}
\newcommand{\PHOJETMC}{PHOJETa,PHOJETb}
\newcommand{\DGLAP}{DGLAPa,DGLAPb,DGLAPc,DGLAPd}

\begin{document}

\pagestyle{empty}
\begin{titlepage}

\noindent
\begin{center}
%{\it {\large version of \today}} \\[.3em] 
\begin{small}
\begin{tabular}{llrr}
%Submitted to & \multicolumn{3}{r}{\footnotesize Electronic Access: {\it http://www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
%Submitted to & \multicolumn{3}{r}{\footnotesize {\it www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline 
Submitted to & & &
\epsfig{file=H1logo_bw_small.epsi,width=2.cm} \\[.2em] \hline
\multicolumn{4}{l}{{\bf
                32nd International Conference 
                on High Energy Physics, ICHEP04},
                August~16,~2004,~Beijing} \\
                 & Abstract:        & {\bf 5-0172}    &\\
                 & Parallel Session & {\bf 5}   &\\ \hline
 & \multicolumn{3}{r}{\footnotesize {\it
    www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em]
\end{tabular}
\end{small}
\end{center}
\vspace*{2cm}


\begin{center}
  \Large
  {\bf 
    Forward Jet Production at HERA}\\
  \vspace*{1cm}
    {\Large H1 Collaboration} 
\end{center}

\begin{abstract} 
\noindent The cross section for inclusive forward jet production
in $ep$ collisions at HERA is presented as a function of $x_{Bj}$,
as is a measurement of the triple differential cross section
$\frac{d^3 \sigma}{dx\, dQ^2\, dp_{t,jet}^2}$ which covers the
phase space from the ``direct photon" region ($Q^2\gg
p_{t,jet}^2$) to the ``resolved virtual photon" region ($Q^2 \ll
p_{t,jet}^2$) via the ``BFKL region" ($Q^2 \sim p_{t,jet}^2$). In
addition, cross sections for events with a central di-jet system
in addition to the forward jet are shown as a function of the
rapidity separation between the forward jet and the two central
jets. The measurements are compared with the predictions of
next-to-leading order QCD calculations and various QCD-based
models; some of these generate parton emissions ordered in
virtuality while others produce non-ordered emissions.

\end{abstract}

\end{titlepage}

\pagestyle{plain}

\section{Introduction}

The hadronic final state in deep inelastic scattering offers an extensive field of 
research for QCD phenomena. This includes studies of hard parton emissions which result
in well defined jets, semi-hard perturbative effects responsible for multiple gluon
emissions and the non-perturbative hadronization process.

HERA has extended the available $\xbj$ region down to values  of $\xbj \simeq
10^{-4} $, for values of  the momentum transfer, $Q^2$,  larger than a few
GeV$^2$, where perturbative calculations in  QCD are still expected to be
valid.  At these low $\xbj$ values, a parton in the proton can induce a QCD
cascade, consisting of several subsequent parton emissions, before eventually
an interaction with  the virtual photon takes place. QCD calculations based on
``direct" interactions between a point-like photon and a parton from an
evolution chain as given by the DGLAP scheme~\cite{dglap}, have been successful
in reproducing  the strong rise of $F_2(\xbj,Q^2)$ with decreasing $\xbj$. On
the other hand, significant  deviations from the simple LO DGLAP approach have
been observed in data on the fractional rate of di-jet events, inclusive jet
production, transverse energy flow and $p_{t,jet}$ spectra of charged
particles. Going from LO to NLO allowed some of the deviations to be resolved,
but in specific regions of phase space the description of the measurements are
still unsatisfactory.

The colour dipole model (CDM)~\cite{cdm}, which assumes that the gluon emission
originates from independently radiating colour dipoles, is in fairly good
agreement with these data. This suggests that new parton dynamics, not included in
the LO DGLAP approach, are responsible for the observed deviations. However,
further investigations made clear that ascribing partonic structure to the
virtual photon and considering so called resolved photon processes is similar
to a full NLO calculation~\cite{kamildijet}. Including leading log parton
showers from both the photon and the proton side leads to a rather satisfactory
description of the data. Thus, more sophisticated measurements are necessary to
establish the existence of new parton dynamics. 

In this analysis we have studied events where a jet has been produced in the
forward direction (the angular region close to the incoming proton), a region
which typically lies well away from the photon end of the evolution ladder. By
applying various cuts  we have tried to suppress DGLAP evolution in order to
become more sensitive to new parton dynamics. Comparisons of data have been
made with next-to-leading order (NLO) calculations and several QCD models. In
this analysis the DISENT program~\cite{disent} has been used to investigate the
level of  agreement between the forward jet data and NLO calculations.

\section{QCD-models} 
At high energies the phase space available for emissions is large. Higher 
order QCD effects will therefore become important and in order to account for
these it is necessary  to use phenomenological models. There are various models
on the market with different approximations to the full evolution equations for
parton branchings, which lead to observable differences in the predictions for
the details of parton cascade.

The most frequently used description so far is given by the DGLAP evolution
equations, which corresponds to the assumption that the leading contribution
comes from strong ordering in the virtualities of the parton propagators in
the evolution chain, with the largest virtualities reached in the hard
scattering with the point-like photon. Compared to the hard scale the
propagator virtualities can be neglected, so that the propagators can be
assumed to be collinear with the incoming proton (collinear approach).
The interaction is assumed to take place with a point-like photon (DGLAP
direct).

In events where the scale of the hard subprocess is larger than $Q^2$, the
structure of the virtual photon might be resolved and the interaction takes
place with one of the partons in the photon. This is described within the 
DGLAP model by introducing two evolution ladders, one from the photon side and
one from  the proton side, and is called the resolved photon
model~\cite{resolved} (DGLAP resolved).

The DGLAP approximation neglects contributions from terms dependent on powers
of $\log(1/x)$, which appear in the full evolution equation, where $x$ is the
longitudinal momentum fraction of the propagating parton. At small enough
$x$-values, these terms eventually dominate over the  $\log(Q^2)$ terms which
are considered by DGLAP. In the small $\xbj$ region, the DGLAP description is
thus expected to break down and the parton dynamics should instead be given by
the BFKL evolution equations~\cite{bfkl}, which resum $\log (1/x)$ terms to all
orders in the coupling constant. This model gives strong ordering in the
longitudinal momentum fraction of the propagators but no ordering in their
virtualities. This means that the virtualities and the  transverse momenta of
the propagators can take any kinematically allowed value at each splitting. One
consequence of this is that the matrix element must be taken off mass-shell and
convoluted with parton distributions which take the transverse momenta of the
propagators into account (unintegrated parton densities).

The CCFM equation~\cite{ccfm} provides a bridge between the DGLAP and BFKL
descriptions by resumming terms in both $\log (Q^2)$ and $\log (1/x)$, which
should make it valid in the complete $x$ range. In the CCFM model the real
emissions are ordered in angle, which gives a correct treatment of colour
coherence effects. The factorization scale is determined by the rescaled
transverse momentum, $q$, of the emitted gluons, which is related to the
maximum angle, $\Xi$, for any emissions at the quark box connecting to the
photon vertex. 

A different approach to the parton evolution is given by the colour dipole
model  in which the emissions are generated by colour dipoles which are spanned
between the partons in the cascade. Since the dipoles radiate independently 
there is no ordering in the transverse momenta of the emissions and the behavior
is thus similar to that of the BFKL case.

\section{Experimental Strategy} 
Differences between the various dynamic approaches to the modelling of the
parton cascade are most prominent in the region close to the proton remnant
direction, i.e. away from the scattered quark. This can be understood from the 
fact that the strong ordering in virtuality of the DGLAP description gives the
softest  emissions closest to the proton whereas in the BFKL model the
emissions can be arbitrarily  hard in this region, as long as they are
kinematically allowed.

\par  
The difficulty in extracting a significant signal for BFKL dynamics is largely
due to the fact that DGLAP parton evolution dominates in most of the HERA
kinematic range. One way to get around this problem is to  select events with a
jet close to the proton direction (a forward jet) with constraints such that
its transverse momentum is approximately equal to the virtuality of the photon
propagator. This will suppress contributions with strong ordering in virtuality
as is the case in DGLAP evolution. Experimentally, this is realized by  
requiring $p_{t,jet}^2 \sim Q^2$, where $p_{t,jet}^2$ is the transverse
momentum squared of the forward jet. If, at the same time, the forward jet is
required to  take a large fraction of the proton momentum,
$x_{jet}=E_{jet}/E_p$, such that $x_{jet} \gg \xbj$, the phase space for an
evolution with ordering in the longitudinal  momentum fraction, as described by
BFKL, is opened up. The $\xbj$ distribution is dominated by the lower limit of
the kinematic acceptance, which is close to $\sim 10^{-4}$.

Based on calculations in the Leading-Log-Approximation of the BFKL kernel, the
cross section for  DIS events at low $\xbj$ and large $Q^2$ with  a forward jet
\cite{fwdjet} is  expected to rise more rapidly with decreasing $\xbj$ than
expected  from DGLAP based calculations.

The analysis presented here is based on a statistical sample which is five
times larger than that used in a previous H1 publication~\cite{prevana} and  is
complementary to a similar analysis~\cite{fwdpion}  which used energetic
forward  pions  instead of  forward jets.  A schematic diagram for forward jet
production is shown in Fig.~\ref{forward_jet_fig}.
\begin{figure}[htb]
  \begin{center} 
    \vspace*{1mm} 
    \vspace*{1cm} 
    \epsfig{figure=fwdjet1.eps,width=8cm,height=8cm} 
    \caption{{\it  
Schematic diagram of an ep scattering event with a forward jet taking a 
fraction $x_{jet}=E_{jet}/E_p$ of the proton momentum. The
evolution from large $\xjet$ to small $\xbj$ is indicated. The phase
space for DGLAP evolution in $Q^2$ is restricted by requiring 
$p_{t,jet}^2 \sim Q^2$.
    \label{forward_jet_fig}}}
  \end{center}
\end{figure}

The requirement of two high transverse momentum central jets in addition to the
forward jet provides further constraints on the kinematics, in the sense that
the virtuality  at the end-points of the gluon ladder is known. The
disadvantage is that this additional requirement gives a significant reduction
of the data sample. Schematic diagrams of such events are shown in
Fig.~\ref{fwdjet-dijet} The position in rapidity space of the two central jets
is relevant to the evolution. A small rapidity separation between the central
jets will leave a large rapidity range for further parton radiation between
the  forward jet and the di-jet system, which is favourable for BFKL
evolution.  In contrast to this, a large rapidity separation between the
central jets means that there is little room for additional emissions between the
central jets and the forward jet. In this case the conditions correspond to
what is expected for resolved photon processes in LO or a 3 parton final state.
\begin{figure}[htb]
  \begin{center} 
    \vspace*{1mm} 
    \vspace*{1cm} 
    \epsfig{figure=fwdjet2.eps,height=5cm} 
    \caption{{\it  
Schematic diagram of an event with a forward jet and a hard di-jet system.
$\eta_i$ denotes the rapidity of the i$^{\rm th}$ jet, $\Delta \eta_1$ the rapidity
difference between the two central jets and $\Delta \eta_2$ the rapidity
difference between the hard subsystem and the forward jet. $x_g$ is the
longitudinal momentum fraction carried by the propagating gluon.
    \label{fwdjet-dijet}}}
  \end{center}
\end{figure}

\section{The H1 Detector}
A detailed description of the H1 detector can be found in~\cite{H1det}. 
The detector elements important for this analysis are described below. The kinematic variables
$\xbj$ and $Q^2$ are determined from a measurement of the scattered electron in the backward
drift chamber (BDC) and the lead-scintillating fibre calorimeter (SPACAL). Jets are
reconstructed using the information provided by the central tracking chambers
and the liquid argon calorimeter (LAr).

Electrons are identified through their energy deposits in the SPACAL
electromagnetic calori- meter and related hits in the BDC. The scattering angle
of the electron can be determined from the reconstructed primary vertex and the
measured impact position in the BDC.  The BDC covers the angular range $153^o<
\theta <177^o$ and gives an accuracy of $\Delta\theta<$ 0.5 mrad.


The SPACAL electromagnetic calorimeter is a lead/scintillating-fibre detector
with a depth of 28 radiation lengths. The energy resolution is $\sigma/E=
7\%/\sqrt{E} \oplus 1\%$, with $E$ in GeV.

The central tracking system has been designed to reconstruct jets with  high
particle densities and to measure the momentum and direction of isolated
charged particles to a precision of  $\sigma_p/p^2 \approx 3 \cdot 10^{-3}$
GeV$^{-1}$ and $\sigma_{\theta} \approx$ 1 mrad. The track reconstruction is
done in two concentric jet chambers (CJC1 and CJC2) with wires oriented
parallel to the beam axis. Two thin drift chambers, one inside CJC1 and one
between the CJC1 and the CJC2,  have their wires perpendicular to the beam
direction and provide the $z$-coordinates  of the tracks. Two proportional
chambers next to the thin drift chambers provide a fast trigger signal for
central tracks.

The main calorimeter is a sandwich type calorimeter with liquid argon as  the
active material. It covers a range in polar angle of $4^o < \theta < 153^o$.
The electromagnetic energy resolution varies between $\sigma/E
=10\%/\sqrt{E}$ and $13\%/\sqrt{E}$ ($E$ in GeV) with a
constant term below $1\%$, whereas the hadronic energy resolution is $\sigma/E
= 50\%/\sqrt{E}$ ($E$ in GeV) with an energy independent term of less than $2\%$.

The luminosity is determined from the measured rate of elastic Bremsstrahlung
scattering,  $e+p\rightarrow e+\gamma+p$ (Bethe-Heitler events), with a luminosity
monitor consisting of two arms. The electron tagger is placed next to the beam
pipe at a distance of  33 m from the nominal interaction point, whereas the
photon detector is situated at 103 m distance in the direction of the incoming
electron beam.  

In this analysis a combination of two triggers is used. Both are based on
energy deposition in the SPACAL detector. The combination of triggers results
in a trigger efficiency of $\sim$100\%.

\section{Event Selection}
The $ep$ scattering data studied here were collected in 1997 at  $\sqrt{s} \sim
300~\GeV$ with the H1 detector and comprise an integrated
luminosity of  $13.7~\pb^{-1}$.

\par
DIS events are selected by requiring  a scattered electron in the backward  SPACAL
calorimeter with an energy $E_e'> 10 $~GeV in the angular range of  $ 156\dg < \theta_e <
175\dg$. The cuts, which are applied in the laboratory frame, are summarized below: 
\begin{center}
\begin{tabular}{c}
$E_{e'} > 10  \mbox{ GeV}$\\
$ 156 \dg < \theta_e < 175\dg$\\
$ 0.1 < y < 0.7$ \\
$ 0.0001 < x_{Bj} < 0.004$ \\
$ 5\mbox{ GeV}^2 < Q^2 < 85 \mbox{ GeV}^2$ .
\end{tabular}§
\end{center}
Here $E_e'$ and $\theta_e$ are the energy and the scattering angle of the
scattered electron, respectively. These variables are determined from the scattered
electron.


The forward jets are defined using the $k_t$-jet 
algorithm~\cite{ktalgo} in its inclusive mode
(applied in the Breit-frame) and by requiring (in the laboratory frame): 
\begin{center}
\begin{tabular}{c}
$p_{t,jet} > 3.5 $ GeV \\
$7.0^o < \theta_{jet} < 20.0^o $  \\
$ \xjet > 0.035 $ .
\end{tabular}\end{center}


\section{Monte Carlo Programs} The H1 data have been compared to the
predictions of several Monte Carlo programs.  The \RAPGAP~\cite{rapgap} Monte
Carlo model (labeled RG) uses LO matrix elements supplemented with initial and
final state DGLAP parton showers for the description of typical DIS-processes
(DGLAP direct). It is interfaced to \HERACLES~\cite{heracles} in order to
simulate QED-radiative effects. \RAPGAP\ also offers the possibility to include
contributions from resolved virtual photon processes (DGLAP resolved). In the
analysis we have used the \DJANGO~\cite{django} program, which provides an
interface to \HERACLES, with the colour dipole model (CDM) as implemented in
\ARIADNE~\cite{ariadne}. The \CASCADE\ Monte Carlo program~\cite{cascade} is
based on the CCFM formalism. Two different versions of the unintegrated gluon
density were used, J2003-set-1 and set-2.  The difference between these two
sets is that in set-1 only singular terms were included in the splitting
function, whereas  set 2 also takes the non-singular terms into account. The
unintegrated gluon densities have been determined from fits to the
$F_2(x,Q^2)$ data obtained by H1 and ZEUS in 1994 and 1996/97~\cite{gludens}.
Simulated events from the RG-DIR and \DJANGO\ Monte Carlo programs have been
processed through the detailed H1 detector simulation in order to test the
understanding of the detector and make acceptance corrections.

A comparison is also made to NLO calculations as obtained by using the \DISENT\
program. These calculations are corrected for hadronization effects, which are
estimated by using CASCADE together with the KMR parton density function. The
KMR parton density function takes only the hard scattering vertex and one
additional emission into account and should therefore be suitable for
correcting the NLO calculations.

\section{Control Plots and Correction Factors} The extent to which the
selection of DIS events and the forward jet sample could be reproduced by the Monte Carlo
programs was investigated through a comparison of data with predictions from the DGLAP-direct model and
the CDM model.  The quality of the DIS selection and the absolute normalization  was
checked by comparing the distributions of DIS events for data and the models  as a
function of the kinematic variables $x_{Bj}$, $y$, $Q^2$ and the energy and polar
angle  of the scattered electron ($E'_e$ and $\theta_e$). Excellent agreement was observed
for both models in all distributions. In Fig.~\ref{dis_dischecks} the distributions of
$x_{Bj}$, $E'_e$ and $\theta_e$ are shown.
\begin{figure}[htb]
  \begin{center} 
    \epsfig{figure=fwdjet3a.eps,height=5.5cm} 
    \epsfig{figure=fwdjet3b.eps,height=5.5cm} 
    \epsfig{figure=fwdjet3c.eps,height=5.5cm} 
    \caption{{\it  
    Control plots for the DIS selection. The distributions are normalized
    according to the luminosity. 
    \label{dis_dischecks}}}
  \end{center}
\end{figure}

For the forward jet selection the distributions of  the jet azimuthal angle
($\phi_{jet}$), the jet rapidity ($\eta_{jet}$), the jet transverse momentum
($p_{t,jet}$), the ratio $p_{t,jet}^2/Q^2$ and the fractional jet energy $x_{jet}=
E_{jet}/E_p$ were examined. The distributions of the DIS  kinematic variables for the
forward jet sample are reproduced better by the CDM model than by DGLAP-direct, whereas
for the forward jet variables the DGLAP-direct model gives a  somewhat better agreement
with data than the CDM model, as shown in Fig.~\ref{fj_fjchecks}
\begin{figure}[htb]
  \begin{center} 
    \epsfig{figure=fwdjet4a.eps,height=5.5cm} 
    \epsfig{figure=fwdjet4b.eps,height=5.5cm} 
    \epsfig{figure=fwdjet4c.eps,height=5.5cm}\\
    \epsfig{figure=fwdjet4d.eps,height=5.5cm} 
    \epsfig{figure=fwdjet4e.eps,height=5.5cm} 
    \epsfig{figure=fwdjet4f.eps,height=5.5cm}
    \caption{{\it  
    Control plots for the forward jet variables, when no $p_{t,jet}^2/Q^2$-cut is applied. The
    distributions are normalized to unity. All variables are measured in the
    laboratory frame. 
    \label{fj_fjchecks}}}
  \end{center}
\end{figure}

The hadron level cross sections were extracted by applying correction factors to the data
which take detector effects into account.  The correction factors were calculated as the
ratio of the Monte Carlo prediction at the hadron- and detector levels, in a bin-by-bin
procedure. RAPGAP and CDM gave very similar values over the full kinematic range covered
in this investigation. The CDM model was finally used to correct data from the detector
level to the hadron level. The correction factors vary between 0.7 and 1.2 but in a few
cases reach 0.5 or 1.4. The variations in the corrections factors from the two Monte
Carlo models are included in the systematic error.  The purity and
acceptance\footnote{The purity (acceptance) is obtained from the same Monte Carlo
simulations as for the correction factors and is defined as the fraction of events
reconstructed (generated) in a bin that were also generated (reconstructed) in
that bin.}
were found to be larger than 30$\%$ in all bins. For the 2+forward jet analysis they are
larger than 40$\%$ in all bins.  

\par
\subsection{Systematic Uncertainties}
The systematic errors have been estimated for each data point separately. In
the figures the quadratic sum of the errors is shown. The total systematic
errors are 10$\%$, 12$\%$ and 14$\%$ for the inclusive, triple and the
2+forward jet cross section, respectively. The following systematic errors are
considered:

\begin{itemize}

\item The energy calibration of the hadronic calorimeter has been performed to
a precision of $\pm 4\%$. In order to estimate the dependence of the measured 
forward jet cross section on this uncertainty, the energy  scale was changed within these
limits and the influence on the forward jet cross section was calculated using the \DJANGO\
generator. The average systematic error is typically $8 \%$ for the inclusive
forward jet cross section and the triple differential forward jet cross section. 

\item For the SPACAL electromagnetic calorimeter the energy scale is known to
an accuracy of $\pm 1\%$. Changing the scale by this amount in the forward jet cross
section calculations results in an average systematic error of typically 3\%.

\item The uncertainty on the measured scattering angle of the electron has been
estimated to be $\pm 1$ mrad. The systematic error which we get by
implementing this measurement uncertainty in the \DJANGO\ forward jet cross section
calculation is typically 1\%.

\item The error from the model dependence has been taken as the difference
between the correction factors calculated from the \DJANGO\ and the RG-DIR
Monte Carlo programs. Taking this variation into account yields a systematic
error of around $5\%$ in the inclusive case and
$8\%$ for the triple differential forward jet cross section. 

\item  The \PHOJET~\cite{phojet} Monte Carlo generator was used in order to
estimate the extent to which DIS forward jet events could be faked by
photoproduction background. The contribution to the forward jet cross section was
calculated to be $\sim 1$\%.

\item The normalization uncertainty of the luminosity measurement has been
estimated to be 1.5\%.

\end{itemize}

\section{Results}
\label{sec:results}
\subsection{Inclusive Forward Jet Production}
\label{sec:inclusive}
The first measurement concerns the inclusive production of forward jets in deep
inelastic scattering. The events were selected by implementing the requirements
described in section 5. Following the discussion in section 3 on how to
suppress the phase space for DGLAP evolution the requirements $0.5 <
p^2_{t,jet}/Q^2 < 5$ and $x_{jet} > 0.035$ GeV were applied. Lowering the upper
$p^2_{t,jet}/Q^2$ cut leads to poor purities. This is caused by the limitations
of the detector resolution in the  $p_{t,jet}$ measurement.

In Fig.~\ref{xfj_djcorr}a the inclusive forward jet cross section is shown 
as a function of $\xbj$ and compared to the prediction of NLO calculations
from DISENT. In Fig.~\ref{xfj_djcorr}b data are compared to the various 
QCD models.
\begin{figure}[htb]
  \begin{center} 
    \vspace*{1mm} 
    \vspace*{1cm} 
    \epsfig{figure=fwdjet5a.eps,height=8cm}(a)
    \epsfig{figure=fwdjet5b.eps,height=8cm}(b)
    \caption{{\it  
    The hadron level cross section for inclusive forward jet production as
a function of $\xbj$ compared to the prediction of (a) NLO calculations and
(b) QCD Monte Carlo models. The band following the data points shows
the uncertainty from variation of the energy scales of the liquid argon calorimeter, the
SPACAL electromagnetic calorimeter and the luminosity monitor. The band
following the NLO calculations illustrates the scale uncertainty in
the calculations, estimated as described in the text.  
    \label{xfj_djcorr}}}
  \end{center}
\end{figure}

The NLO calculations were performed using the CTEQ6M parametrization of the
proton parton densities with the renormalization scale given by the $E_T^2$ of the jet.
The average $E_T^2$ of the jets (45 and 67 GeV$^2$ for the inclusive and the
triple differential cross sections, respectively) was used as the factorization scale.
The scale uncertainty was estimated by changing the scale by a factor of four,  
$E_T^2/4<\mu_r^2<4E_T^2$, and is indicated as a band in the plot. 
The parametrization of the parton densities and the scale used in the 
QCD models are given in table~\ref{generatortable}.
  
From the figures it is obvious that the DGLAP model with direct photon
interactions alone (RG-DIR) and the NLO calculation both fall below the data:
This is especially pronounced at low $x_{Bj}$. The somewhat improved agreement
at higher $x_{Bj}$ can be understood from the fact that  the range in the
longitudinal momentum fraction which is available for higher order emissions is
decreased due to the $x_{jet}$ cut. The description of the data by the
DGLAP-model is significantly improved if contributions from resolved virtual
photon interactions are included (RG-DIR+RES). However, there is still a
discrepancy in the lowest  $x_{Bj}$-bin, where a possible BFKL signal would be
expected to show up most prominently. The CDM model, which gives emissions that
are non-ordered in transverse momentum, shows a similar behaviour to the RG
DIR+RES model.  In addition the CCFM-model (with both set-1 and set-2 partons)
predicts a somewhat different shape for the  $x_{Bj}$ distribution, which
results in a comparatively poor description of  the data. 

\begin{center}
\begin{table}[h]
\begin{tabular}{|c||c|c|c|c|}
\hline
 & \CASCADE\ & RG-DIR & RG-RES & \DISENT\ \\
\hline
\hline
$\mu_r^2$ & $m^2+p_{t,jet}^2$ & $Q^2+p_{t,jet}^2$ & $Q^2+p_{t,jet}^2$ & $\frac{1}{4}p_{t,jet}^2 < p_{t,jet}^2 
< 4p_{t,jet}^2$ \\
\hline
$\mu_f^2$ & Determined by $\Xi$ & $Q^2+p_{t,jet}^2$ & $Q^2+p_{t,jet}^2$ &
$<p_{t,jet}^2> (=45 \textrm{ resp. } 67 \textrm{ GeV}^2)$\\
 \hline
proton pdf  & J2003 set 1 $\&$ 2 & CTEQ6L & CTEQ6L & CTEQ6M\\
\hline
photon pdf  & - &  - & SaS1D & -\\
\hline
\end{tabular}
\caption{{\it Scales and parton density functions used in the different 
generators. $\Xi$ denotes the maximum emission angle given by the quark box determining the factorization scale. 
\label{generatortable}}}
\end{table}
\end{center}

\subsection{Triple Differential Cross Sections}  
\label{sec:triple}

In order to get a more complete picture of forward jet production the data are
also presented as triple differential cross sections. The total forward jet
event sample was subdivided into bins of $Q^2$ and  $p_{t,jet}^2$. The triple
differential cross section  $\frac{d\sigma}{dx_{Bj}dQ^2dp_{t,jet}^2}$ versus
$x_{Bj}$ is shown in Figs.~\ref{xp2q2hadnlo} and~\ref{xp2q2had}  for three
regions in $Q^2$ and $p_{t,jet}^2$. In Fig.~\ref{xp2q2hadnlo} the data are
compared to  the DISENT NLO calculations, whereas in Fig.~\ref{xp2q2had}
comparisons to the Monte Carlo models are shown. The same parton density
functions and scales have been used as in the inclusive case. Again the scale
uncertainty is represented by a band in Fig.~\ref{xp2q2hadnlo}.
\begin{figure}[htb!]
  \begin{center} 
    \vspace*{1mm} 
    \vspace*{1cm} 
    \epsfig{figure=fwdjet6.eps,width=15cm,height=17.5cm}
        \caption{{\it  
    The hadron level triple differential cross section for forward jet
production as a function of $\xbj$, in bins of $Q^2$ and $p_{t,jet}^2$. The
data are compared to the prediction  of NLO calculations. The band following
the data points illustrates the uncertainty due to variation in the energy
scales of the liquid argon calorimeter, the SPACAL electromagnetic calorimeter
and the luminosity monitor. The band following the NLO calculations illustrates the scale
uncertainty in the calculations.  In every bin the coverage in
$r=p_{t,jet}^2/Q^2$ is shown.
    \label{xp2q2hadnlo}}}
  \end{center}
\end{figure}
\begin{figure}[htb]
  \begin{center} 
    \vspace*{1mm} 
    \vspace*{1cm} 
    \epsfig{figure=fwdjet7.eps,width=15cm,height=17.5cm}
        \caption{{\it  
    The hadron level triple differential cross section for forward jet production as
a function of $\xbj$, in bins of $Q^2$ and $p_{t,jet}^2$. The data are compared to the prediction 
of QCD Monte Carlo models. The band following the data points illustrates 
the uncertainty in the energy scales of the liquid argon calorimeter, the
SPACAL electromagnetic calorimeter and the luminosity monitor. In every bin the coverage
in $r=p_{t,jet}^2/Q^2$ is shown. RG-DIR+RES is here denoted by RGtot.
    \label{xp2q2had}}}
  \end{center}
\end{figure}
  
From Fig.~\ref{xp2q2hadnlo} it can be observed that the NLO calculations, within the
fairly large scale uncertainty, agree with the data in the regions of high $Q^2$ and/or high
$p_{t,jet}^2$. For lower values of these parameters the NLO calculations fall below the
data. This is consistent with the results from a previous measurement on inclusive jet
production~\cite{desy-02-079}.
  
The kinematic region covered in Figs.~\ref{xp2q2hadnlo}c and~\ref{xp2q2had}c 
includes the case where $Q^2$ 
is larger than $p_{t,jet}^2$, which is typical for direct photon interactions,
but it extends into the region where $Q^2$ is approximately equal to
or even smaller than $p_{t,jet}^2$, and where emissions non-ordered in virtuality are expected.
This could explain why the DGLAP direct model (RG-DIR) does not give
a good description of the data except for the highest $x_{Bj}$-bin. 
The CDM model reproduces the data very well
and the DGLAP resolved model (RG-DIR+RES) is also in reasonable agreement. 
The CCFM model (CAS1 and CAS2) overshoots the data over the full $x_{Bj}$-region.

Figs.~\ref{xp2q2hadnlo}d, e, g, h and i, and ~\ref{xp2q2had}d, e, g, h and i cover a 
kinematic region where $p_{t,jet}^2$
is larger than $Q^2$, which is typical for processes where the 
virtual photon is resolved. 
As expected the DGLAP resolved model (RG-DIR-RES) a good overall good description of the data, whereas 
DGLAP direct (RG-DIR) and NLO calculations give cross sections which are generally 
too low. The CDM-model overshoots the data significantly 
at high values of $r = \frac{p_{t,jet}^2}{Q^2}$ and small values of $x_{Bj}$. 

The ``BFKL region", with $p_{t,jet}^2$ of the same order as $Q^2$, is
represented by Figs.~\ref{xp2q2hadnlo}b and f, and ~\ref{xp2q2had}b and f,
where Figs.~\ref{xp2q2hadnlo}b and ~\ref{xp2q2had}b cover the lower range in
$p_{t,jet}^2$ and $Q^2$ and Figs.~\ref{xp2q2hadnlo}f and  ~\ref{xp2q2had}f the
higher range. In this kinematic region the data are best described by the DGLAP
resolved (RG-DIR+RES) model, whereas the CDM model gives somewhat too low
cross-sections. In the bin of low $p_{t,jet}^2$ and $Q^2$
(Fig.~\ref{xp2q2had}b) there is a tendency for CCFM (CAS1 and CAS2)  to
underestimate the production cross section at low values of $x_{Bj}$ and
overestimate it at higher values, as was already observed in the inclusive
distribution.  For low values of $p_{t,jet}^2$ and high $Q^2$
(Fig.~\ref{xp2q2had}c, f)  the CCFM predictions are significantly too high over
the full $x_{Bj}$-range.



\subsection{Events with Reconstructed Di-jets in Addition to the Forward Jet}

Complementary to the analysis reported in section~\ref{sec:inclusive}
and~\ref{sec:triple}, where the $p_{t,jet}^2/Q^2$ cut was used to enhance a
possible BFKL signal, we also used another method to control the evolution
kinematics. By requiring the reconstruction of two jets in the central region of
the detector, we can investigate different kinematic regions by applying
cuts on the jet momenta and their rapidity separation as described in more
detail in section 3.

Di-jets from the central region were found by applying the inclusive $k_t$-jet
algorithm in the Breit frame and demanding that jets have transverse momenta
larger than 6 GeV. The jets are ordered  in rapidity according to
$\eta_{fwd-jet} > \eta_{jet_2} > \eta_{jet_1} > \eta_e$  with $\eta_e$ being
the rapidity of the scattered electron, see Fig.~\ref{fwdjet-dijet}. The
cross section is  measured in two intervals of $\Delta \eta_1 =  \eta_{jet_2} -
\eta_{jet_1} $. By applying the same $p_{t,jet}$ cut for all three jets,
evolution with strong  $k_{t}$-ordering is suppressed. In order to maximize the
phase space available for BFKL evolution $\Delta \eta_1 < 1$ was required. This
means that the invariant mass of the  di-jet system and $x_g$ are small (see
Fig.~\ref{fwdjet-dijet}). A consequence of this is that the rapidity
difference of the di-jet system to the forward jet is maximized.  On the other
hand, by demanding  $\Delta \eta_1 > 1$ we select di-jet systems with  higher
invariant masses and larger $x_g$; the  separation of the di-jet system and the
forward jet becomes smaller and a  description corresponding to the resolved
photon picture should become adequate. In this investigation no comparison with
NLO($\alpha_s^2$)-calculations is made since these by construction are
limited to the production of three jets. The same versions of the QCD models
were used as in the previous studies. 

The cross section for events containing a di-jet system in addition to the
forward jet is shown in Fig.~\ref{2+fwdjet}, as a function of  $\Delta \eta_2 =
\eta_{fwd-jet}-\eta_{jet_2}$ for all jets, and for the requirements $\Delta
\eta_1<1$ and $>1$.  The cross section $d\sigma/d\Delta\eta_2$ for
$\Delta\eta_1<1$ is found to be fairly well described by the \CASCADE\ MC with
set-2 parton densities except in the lowest bin where  all models fail. This is
the bin where the rapidity range for additional radiation  is the smallest.
\CASCADE\ with set-1 partons, however, gives too high cross section values in
the two highest $\Delta \eta_2$ bins. The DGLAP direct and resolved models are
both significantly below the data in the whole range. In the sample where
$\Delta\eta_1>1$, the data are not described by any model in the lowest
$\Delta\eta_2$-bin, while for the two other bins both \CASCADE\ set-1 and set-2
give cross sections that are too large. The DGLAP models give good agreement in
the highest $\Delta\eta_2$-bin. The observed behaviour is consistent with that
expected from the discussion above. Within the uncertainty in the energy
scale, the CDM model gives somewhat better agreement with data for
$\Delta\eta_1<1$ (the BFKL region) than for $\Delta\eta_1>1$ (the resolved
region), where it fails to reproduce the lowest $\Delta\eta_2$ bin.

\begin{figure}[htb]
  \begin{center} 
    \vspace*{1mm} 
    \vspace*{1cm} 
    \epsfig{figure=fwdjet8a.eps,width=5.2cm}
    \epsfig{figure=fwdjet8b.eps,width=5.2cm}
    \epsfig{figure=fwdjet8c.eps,width=5.2cm}
                \caption{{\it  
    The cross section for events with a reconstructed high transverse momentum
    central di-jet system and a forward
jet as a function of the rapidity gap between the forward jet and the most
forward-going central jet, $\Delta\eta_2$. Results are shown for the full sample and
for two ranges of the separation between the two central jets, $\Delta\eta_1<1$ and $\Delta\eta_1>1$.
The data are compared to the predictions of QCD Monte Carlo models.
    \label{2+fwdjet}}}
  \end{center}
\end{figure}



\section{Conclusion}

An investigation of DIS events containing a jet in the forward direction has
been performed using data collected in 1997, comprising an integrated 
luminosity of $13.72$ pb$^{-1}$. Various constraints have been applied which
suppress contributions to the parton evolution described by the DGLAP equations
and thus enhance the sensitivity to new  parton dynamics. Several
observables involving forward jet events have been studied  and compared to the
predictions of NLO calculations and QCD models.

The results on inclusive forward jet production show that NLO calculations and
the DGLAP direct model give cross sections which are consistently below  the
data at small values of $\xbj$. The DGLAP resolved photon model and colour
dipole model give the best description of the data, whereas the CCFM model,
studied with two different parametrizations of the unintegrated gluon density,
does not reproduce the shape of the distribution. This shows that the forward
jet cross section is sensitive to the details of the unintegrated gluon density
and can be used to further constrain this density.

The total forward jet sample was subdivided into bins of $Q^2$ and
$p_{t,jet}^2$ such that kinematic regions were defined in which different
evolution dynamics were expected to dominate. At high $Q^2$, ($Q^2 \gg
p_{t,jet}^2$) the most DGLAP like region, the data are described by NLO
calculations within the scale uncertainty. In the region where contributions
from resolved processes are expected to become important ($p_{t,jet}^2 \gg
Q^2$) we find good  agreement with the DGLAP resolved model but the cross
sections predicted by the NLO calculations  and DGLAP direct model are too low.
In this region the CDM tends to overshoot the data. In the BFKL region ($Q^2
\sim p_{t,jet}^2$) the CCFM model does not manage to describe the shape of the
distributions and CDM and DGLAP resolved reproduce the data best.

The study of events with a reconstructed central di-jet system in addition to the
forward jet reveals reasonably good agreement with CASCADE in the region where BFKL
evolution is expected to dominate. In the region where we expect resolved photon
processes to become important the DGLAP resolved model is closer to the data 

The observations made here demonstrate that an accurate description of the
radiation pattern at small $x_{Bj}$ requires the introduction of terms
beyond those present in the collinear DGLAP approximation. Higher order
parton emissions with significant transverse momentum contribute noticeably
to the cross section. Calculations which include these processes, such as
CCFM, CDM and the resolved photon model, provide a better description of the
data.

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\end{document}


